Simon has $160$ meters of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width $x$ (in meters) is modeled by $A(x)=-x(x-80)$ What width will produce the maximum garden area?
The garden's area is modeled by a quadratic function, whose graph is a parabola. The maximum area is reached at the vertex. So in order to find the width that produces the maximum area, we need to find the vertex's $x$ -coordinate. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} A(x)&=0 \\\\ -x(x-80)&=0 \\\\ \swarrow &\searrow \\\\ -x=0\text{ or }&x-80=0 \\\\ x={0}\text{ or }&x={80} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({0})+({80})}{2}=\dfrac{80}{2}=40$ In conclusion, the maximum area is produced when the rectangle's width is $40$ meters.